A horizon is a limit on causal access within a spacetime model. Several different limits occur in cosmology, and they should not be replaced by one imagined edge of the universe. A small family of expansion histories makes their differences calculable.
Specify the history before naming its horizons
Use a spatially flat FLRW model with a(t) = (t/t₀)p, positive t and p, and a(t₀) = 1. Assume this law applies from t approaching zero into an indefinitely long future. That strong assumption is what makes the horizon integrals below definite. None of these curves is a fit to the full observed expansion history.
For a comoving source at redshift z, put u = tₑ/t₀ = (1 + z)−1/p. The lookback time in units of t₀ is 1 − u. Its present proper distance in units of ct₀ is the integral of v−p from u to 1.
For p ≠ 1 this distance is [1 − u1−p]/(1 − p); for p = 1 it is −ln(u). The difference from 1 − u reflects the changing geometry during light propagation. Present proper distance is assigned on today's spatial slice; it is not a direct ruler laid along the photon's entire journey.
Compare distance, lookback and causal limits
POWER-LAW FLRW MODEL · DISTANCES IN ct₀
Solid green: present distance. Dashed gold: c × lookback time. Both are divided by ct₀; dots mark the selected redshift.
| Quantity today | Distance / ct₀ | Meaning in this model |
|---|---|---|
| Hubble radius | 1.5000 | Instantaneous scale c/H₀ |
| Particle horizon | 3.0000 | Finite past light-travel bound |
| Event horizon | No finite horizon | Future integral diverges |
At p = 2/3 and z = 3, present distance equals the Hubble radius, while the particle horizon is twice as far away.
Three quantities answer three questions
The Hubble radius is c/H₀ = ct₀/p. It is an instantaneous scale defined by the expansion rate. It does not generally mark the boundary of the observable universe. Davis and Lineweaver's published analysis explains why observations of sufficiently distant receding sources do not violate the local speed limit.
The particle horizon uses the past integral from zero to t₀. Here it has present proper radius ct₀/(1 − p) when p < 1. When p ≥ 1 that integral diverges: the ideal model has no finite particle horizon. This statement concerns the extrapolated history, not the sensitivity of a telescope.
The event horizon uses the future integral from t₀ to infinity. It has present proper radius ct₀/(p − 1) when p > 1; otherwise there is no finite event horizon. Its existence therefore depends on the future assumed by the model. Changing the future expansion can change this conclusion without changing a photograph taken today.
Read two contrasting cases
At p = 2/3 and z = 3, emission occurred at one eighth of the model's current age. Light travelled for seven eighths of that age, yet the present distance is 1.5ct₀. The particle horizon is 3ct₀. These three numbers cannot consistently be described as one “distance light has travelled.”
Switch to p = 2. The model accelerates, has a finite event horizon of ct₀, and has no finite particle horizon under its assumed past extension. Past visibility and future communication can therefore have very different limits.
Finite opacity, faintness and foreground contamination impose additional observational limits. The microwave last-scattering surface is not simply identical to a particle horizon. A causal opportunity for a signal and a usable observable signal are separate requirements.
Shared origin must still recover causal access
A common generative source in SMM/DTT does not by itself specify which completed observers can exchange records. The book's requirements for gravitation include a metric and its dynamics. Those ingredients are needed to calculate the light paths and limits explored here.
Sources and further reading
- T. M. Davis and C. H. Lineweaver, Expanding Confusion, Publications of the Astronomical Society of Australia 21, 97–109 (2004).
- David Tong, The Expanding Universe, for FLRW geometry and horizon definitions. The interactive calculation is an original teaching example.
- Continue with light cones and boundaries of cosmological models.